Pre-1968 children's book about a stray dog fed by multiple families

Hi there, I'm trying to help my mom track down a children's book from her childhood. She was born in 1963 and remembers the book from age 5, so it had to have been published before 1968. Per her description, the book is about a dachshund/wiener dog that gets fed by multiple neighbors, who all think that the dog is theirs. The families catch on to the fact that he's visiting other houses when they realize he's getting fat. The story resolves by the neighbors putting up a big fence that encircles all their backyards, so the dog can come and go as he pleases. Thank you in advance for any leads!